Measuring the Surface- Tension of Liquids, 405 



and putting for cos a and sin a their values from (2), the 

 condition that w shall be a maximum is 



16?^+12c 2 ry d + 2c\e 2 -Sr 2 )y-Sc 4 ry-c G = 0. . (7) 



The numerical value of y may be found in any particular 



7T 



case by assuming that when w is a maximum u = - , and 



finding for c 2 from (4) an approximate value to use in (7) . 



h is a maximum when dh/dw = 0, in which case a = ir and 

 w = 7rr 2 p. But before this condition is attained the liquid 

 surfaces meet below the cylinder and form (in some cases) a 

 film. This occurs when (fig. 6) 



+ %= +r sin «, 



= ± 2 ^V^?,from(2), 



= £ i og c±>Jc-y _ i/cW+ -51572 c, from (3), 

 2 & y 



or 



I6 g c - ^-y = (H + gy-^p-l-03144, . (8) 



y \ G c J 



from which y may be found approximately in terms of c and 

 r by trial. The required value of y is greater than c/s/2 

 and less than c, hence the selection of signs in (8). 



As soon as a true film is formed having plane surfaces, the 

 quantity 2ry sin a in (4) is reduced to 2r sin «(?/— c/x/2), 

 because the column of liquid below the film no longer acts 

 directly to produce a (negative) pressure upon the bottom of 

 the cylinder ; and w is reduced to 2T plus the weight of the 

 cylinder and of the small quantity of liquid suspended under 

 the cylinder and above the film. 



Up to this point the ends of the cylinder, which in practice 

 must be turned down, have been disregarded. Their effect 

 may be eliminated by the use of a long and a short cylinder 

 having the same thickness and their ends alike ; but if the 

 bend is like an elbow, or is made by a rectangular prism 

 abutting squarely on the end of the cylinder, the effect of the 

 ends may be calculated. 



Taking the former case, let I (fig. 7) be the whole length 

 of the cylinder along the dotted line, r its radius, h as before 

 the height of the top of the horizontal cylinder above the 

 general level of the liquid, and k the length of each vertical 

 end-piece, measured from the top. Equation (4) may be 

 applied over the length l — 2r — 7rr/2. And if we subtract 



